Dynamical systems are mathematical models characterized by a set of differential or difference equations. Model reduction aims to replace the original system with a reduced system of significantly smaller dimension that still describes the important dynamics of the large-scale model. Interpolatory model reduction methods define a reduced model that interpolates the full model at selected interpolation points. The reduced model may be obtained through a Krylov reduction process or by using the Iterative Rational Krylov Algorithm (IRKA), which iterates this Krylov reduction process to obtain an optimal ℋ₂ reduced model. This dissertation studies interpolatory model reduction for first-order descriptor systems, second-order systems, and DAEs. The main computational cost of interpolatory model reduction is the associated linear systems. Especially in the large-scale setting, inexact solves become desirable if not necessary. With the introduction of inexact solutions, however, exact interpolation no longer holds. While the effect of this loss of interpolation has previously been studied, we extend the discussion to the preconditioned case. Then we utilize IRKA's convergence behavior to develop preconditioner updates. We also consider the interpolatory framework for DAEs and second-order systems. While interpolation results still hold, the singularity associated with the DAE often results in unbounded model reduction errors. Therefore, we present a theorem that guarantees interpolation and a bounded model reduction error. Since this theorem relies on expensive projectors, we demonstrate how interpolation can be achieved without explicitly computing the projectors for index-1 and Hessenberg index-2 DAEs. Finally, we study reduction techniques for second-order systems. Many of the existing methods for second-order systems rely on the model's associated first-order system, which results in computations of a 2𝑛 system. As a result, we present an IRKA framework for the reduction of second-order systems that does not involve the associated 2𝑛 system. The resulting algorithm is shown to be effective for several dynamical systems. / Ph. D.

The attitude (or orientation) of an object is often crucial in its ability to perform a task, whether the task is driving a car, flying an aircraft, or focusing a satellite. In traditional control approaches, the attitude is often parameterized by Euler angles or unit quaternions which exhibit problems such as gimbal lock or ambiguity in representation, respectively. These complications prevent the controllers from achieving global stability and worse they may cause real physical harm due to unexpected large motions. More recent works have achieved global stability and avoided these system failures by working directly on the configuration manifold, but these approaches are generally complex or lack automatic, user-friendly ways to tune them. The goal of this dissertation is to develop simple geometric attitude controllers that are globally, exponentially stable and can be automatically tuned. By simple, we mean that the controllers are computationally efficient for real time implementation on embedded computers and the tuning parameters have geometric interpretations. These properties make the controllers user friendly and practical for real hardware implementation even on fast dynamical systems. Furthermore, we aim to obtain an automatic tuning procedure that ensures convergence, and can also quantify and optimize performance guarantees. We achieve our goal through four major contributions. The first is a substantial generalization on the theory of classical Riemannian metrics for tangent bundles which provides the ability to compare and combine attitude and velocity terms in the stability analysis, allowing us to consider a larger set of feasible controller gains. The second contribution is a framework to study the stability of attitude systems on manifolds and to automatically tune the controller gains by combining Riemannian geometry, contraction theory, and offline optimization. The third contribution is the development of a globally, exponentially stable attitude controller. This controller overcomes the topological limitation that prevents continuous, time-invariant controllers from achieving global stability by using a time-varying intermediate reference trajectory. The fourth contribution is the improvement of the proposed controllers by way of point-wise-in-time quadratic programming.